论文标题
符合理性表面的符合性摩尔群
Symplectic Torelli groups of rational surfaces
论文作者
论文摘要
如果$ c_1（x）\ cdot [ω]> 0 $，我们称之为符号有理表面$（x，ω）$ \ textit {pastic}。理性表面的阳性条件相当于存在$ d \ subset x $的存在，因此$（x，d）$是log calabi-yau表面。符合形式的同时类别可以使用与Lagrangian球形类关联的root System赋予\ textit {type}。在本文中，我们证明，如果是$ \ mathbb {a} $的类型，那么正理性表面的符号托雷利组是微不足道的，如果它是类型$ \ mathbb {d} $，则是一个球体编织组。 作为一个应用程序，我们肯定地回答了一个长期的公开问题，即拉格朗尼亚球形dehn Twist会产生Symplectic Torelli组$ symp_h（x）$时，$ x $是正理性的。我们还证明，所有符号曲面表面都有微不足道的象征性摩尔氏群体。最后，我们验证了清 - 凯斯勒的符合性交流是哈密顿人，他积极回答了凯德拉的问题。 我们的主要新投入是最近由于Li-Zhang和Zhang引起的几乎复杂的亚地区的研究。受这些作品的启发，我们为正理性表面的几乎复杂结构定义了一个新的\ textit {粗分层}。我们还将符号磁场理论和用于应用程序的参数化gromov-witten理论结合在一起。
We call a symplectic rational surface $(X,ω)$ \textit{positive} if $c_1(X)\cdot[ω]>0$. The positivity condition of a rational surface is equivalent to the existence of a divisor $D\subset X$, such that $(X, D)$ is a log Calabi-Yau surface. The cohomology class of a symplectic form can be endowed with a \textit{type} using the root system associated to its Lagrangian spherical classes. In this paper, we prove that the symplectic Torelli group of a positive rational surface is trivial if it is of type $\mathbb{A}$, and is a sphere braid group if it is of type $\mathbb{D}$. As an application, we answer affirmatively a long-term open question that Lagrangian spherical Dehn twists generate the symplectic Torelli group $Symp_h(X)$ when $X$ is a positive rational surface. We also prove that all symplectic toric surfaces have trivial symplectic Torelli groups. Lastly, we verify that Chiang-Kessler's symplectic involution is Hamiltonian, answering a question of Kedra positively. Our key new input is the recent study of almost complex subvarieties due to Li-Zhang and Zhang. Inspired by these works, we define a new \textit{coarse stratification} for the almost complex structures for positive rational surfaces. We also combined symplectic field theory and the parametrized Gromov-Witten theories for our applications.
